Rendleman–Bartter model

In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomorphic. The theorem is important in the associated bundle construction where one starts with a given bundle and surgically replaces the fiber with a new space while keeping all other data the same.

Formal statement

Let X and F be topological spaces and let G be a topological group with a continuous left action on F. Given an open cover {U_i} of X and a set of continuous functions

$t_{ij}:U_i\cap U_j\to G$

defined on each nonempty overlap, such that the cocycle condition

$t_{ik}(x)=t_{ij}(x)t_{jk}(x)\mathrm{\forall }x\in U_i\cap U_j\cap U_k$

holds, there exists a fiber bundle E → X with fiber F and structure group G that is trivializable over {U_i} with transition functions t_ij.

Let E′ be another fiber bundle with the same base space, fiber, structure group, and trivializing neighborhoods, but transition functions t′_ij. If the action of G on F is faithful, then E′ and E are isomorphic if and only if there exist functions

$t_i:U_i\to G$

such that

$t{\text{'}}_{ij}(x)=t_i(x{)}^{-1}{t}_{ij}(x)t_j(x)\mathrm{\forall }x\in U_i\cap U_j.$

Taking t_i to be constant functions to the identity in G, we see that two fiber bundles with the same base, fiber, structure group, trivializing neighborhoods, and transition functions are isomorphic.

A similar theorem holds in the smooth category, where X and Y are smooth manifolds, G is a Lie group with a smooth left action on Y and the maps t_ij are all smooth.

Construction

The proof of the theorem is constructive. That is, it actually constructs a fiber bundle with the given properties. One starts by taking the disjoint union of the product spaces U_i × F

$T=\coprod _{i\in I}{U}_i\times F=\{(i,x,y):i\in I,x\in U_i,y\in F\}$

and then forms the quotient by the equivalence relation

$(j,x,y)\sim (i,x,t_{ij}(x)\cdot y)\mathrm{\forall }x\in U_i\cap U_j,y\in F.$

The total space E of the bundle is T/~ and the projection π : E → X is the map which sends the equivalence class of (i, x, y) to x. The local trivializations

${\varphi }_i:{\pi }^{-1}(U_i)\to U_i\times F$

are then defined by

${\varphi }_i^{-1}(x,y)=[(i,x,y)].$

Associated bundle

Let E → X a fiber bundle with fiber F and structure group G, and let F′ be another left G-space. One can form an associated bundle E′ → X a with fiber F′ and structure group G by taking any local trivialization of E and replacing F by F′ in the construction theorem. If one takes F′ to be G with the action of left multiplication then one obtains the associated principal bundle.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 See Part I, §2.10 and §3.